The present invention is broadly applicable in the rapidly developing field of quantum computing. Research in the field of quantum computing began in 1982 when physicist Richard Feynman introduced the concept of a “quantum simulator.” See R. P. Feynman, “Simulating Physics with Computers”, Int. J. Theor. Phys., 21:467-488 (1982). Feynman noted that quantum systems are inherently difficult to simulate with conventional computers but that observing the evolution of an analogous quantum system could provide an exponentially faster way to solve the mathematical model of a system. In particular, solving a model for the behavior of a quantum system commonly involves solving a differential equation related to the Hamiltonian of the quantum system. Soon it was determined that a quantum system could be used to yield a potentially exponential time saving in certain types of intensive computations. See D. Deutsch, “Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer”, Proc. of the Roy. Soc. of London A400:97-117 (1985).
Since then, research has progressed to include significant software and hardware advances. As the speed of classical computers approaches a projected upper bound due to the natural limits of miniaturization of integrated circuits, so interest in quantum computers has intensified. Indeed, many algorithms have been written to run on quantum computers, two notable examples being the Shor and Grover algorithms. See P. Shor, “Polynomial-time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer”, SIAM J. of Comput., 26(5): 1484-1509 (1997), and U.S. Pat. No. 6,317,766 entitled “Fast Quantum Mechanical Algorithms”; and L. Grover, “A Fast Quantum Mechanical Algorithm for Database Search”, Proc. 28th STOC, 212-219 (1996). It has further been shown that preventive error correction schemes for quantum computing are theoretically possible. See U.S. Pat. No. 5,768,297, entitled: “Method for reducing decoherence in quantum computer memory” to P. W. Shor; and U.S. Pat. No. 6,128,764, entitled: “Quantum error-correcting codes and devices”, to D. Gottesman.
Furthermore, methods of operating quantum computers have been described. See, for example, in U.S. Pat. No. 6,301,029, entitled: “Method and apparatus for configuring quantum mechanical state, and communication method and apparatus using the same”, to H. Azuma, and in U.S. Pat. No. 6,317,766, entitled: “Fast quantum mechanical algorithms”, to L. Grover. These high level algorithms have relevance to proposals for solid state implementations.
Thus, the quantum computer is now rapidly evolving from a wholly theoretical idea to a physical device that will have a profound impact on the computing of tomorrow. A quantum computer differs principally from a conventional, semiconductor chip-based computer, in that the basic element of storage is a “quantum bit”, or “qubit”. Generally speaking, a qubit is a well-defined physical structure that has a plurality of quantum states, that can be isolated from its environment and that can evolve in a quantum mechanical fashion. A survey of the current physical systems from which qubits could be formed can be found in: S. L. Braunstein and H. K. Lo (eds.), Scalable Quantum Computers, Wiley-VCH Verlag GmbH, Berlin (2001), incorporated herein by reference. A qubit is a creature of the quantum world: it can exist in a superposition of two states and can thereby hold more information than the binary bit that underpins conventional computing. One of the principal challenges in quantum computing is to establish an array of controllable qubits, so that large scale computing operations can be carried out.
Although a number of different types of qubits have been created, it is believed that the practical realization of a large scale quantum computer is most likely to be achieved by harnessing the properties of superconducting junctions. It is in the superconducting regime that many materials display their underlying quantum behavior macroscopically, thereby offering the chance for manipulation of quantum states in a measurable way. For example, a superconducting qubit and register, i.e., an array of interacting qubits, has been described in U.S. Pat. No. 6,459,097 B1 entitled “Qubit using a Josephson Junction between s-Wave and d-Wave Superconductors,” to Zagoskin. The field has been reviewed, for example, by Y. Makhlin, G. Schön, and A. Shnirman in “Quantum-State Engineering with Josephson-Junction Devices”, Reviews of Modern Physics, Vol. 73, pp. 357-400 (2001).
The early theoretical work in the field of quantum computing led to the creation of a formal theory of the nominal capabilities that are held to be necessary conditions for a physical system to behave as a qubit, and for a series of qubits to become a quantum computer. See D. DiVincenzo in Scalable Quantum Computers, S. L. Braunstein and H. K. Lo (eds.), chapter 1, Wiley-VCH Verlag GmbH, Berlin (2001), also published as ArXiv preprint “quant-ph/0002077” (2000), incorporated herein by reference. These requirements include the need for the system to be scalable, i.e., the ability of the system to combine a reasonable number of qubits. Associated with scalability is the need to eliminate decoherence in a qubit. Also required for a qubit to be useful in quantum computing is the ability to perform operations that initialize, control and couple qubits. Control of a qubit includes performing single qubit operations as well as operations on. two or more qubits. Coupling is an operation performed on two or more qubits. Finally, it is necessary to be able to measure the state of a qubit in order to perform computing operations.
To make a practical design for a quantum computer, one must specify how to decompose any valid quantum computation into a sequence of elementary one qubit and two qubit quantum gates that can be realized in physical hardware that is feasible to fabricate. The set of these one and two qubit gates is arbitrary provided it is universal, i.e., capable of achieving any valid quantum computation from a quantum circuit comprising only gates from this set. The set of gates also needs to be a universal set, i.e., one which permits universal quantum computation. Fortunately, many sets of gates are universal, see A. Barenco et al, “Elementary Quantum Gates for Quantum Computation”, Physical Review A 52:3457 (1995), and references therein, which is incorporated herein by reference.
A widely accepted method of operating quantum computers is the “standard paradigm” of universal quantum computation. According to the standard paradigm, all operations necessary for a quantum computer can be performed by single qubit and two qubit operations, because these two types of operations generate the full special unitary 2 group, denoted SU(2N), which spans the space necessary for quantum computation. Quantum computers that generate the full SU(2N) group space for N qubits are sometimes referred to as universal quantum computers. In particular, two single qubit gates that are based on two non-commuting Hermitian operators can generate all one qubit quantum gates, and a two qubit gate can entangle the states of a two qubit quantum system. One qubit quantum gates alone, however, are only sufficient to generate the SU(2) group, which only spans the space necessary for quantum computation with a single qubit.
The two principal single qubit operations are tunneling and biasing. Typical designs, for example, that of U.S. application Ser. No. 09/839,637 require single qubit biasing. The operation of the superconducting elements and the basic devices has also been described, for example, by A. Blais and A. M. Zagoskin in “Operation of universal gates in a solid-state quantum computer based on clean Josephson junctions between d-wave superconductors,” Physical Review A, Vol. 61, 042308 (2000), available as ArXiv.org/abs/quant-ph/9905043. This work described a system with single qubit bias, through circuits and other methods. However, designs with single qubit bias add considerable complication and expense to the fabrication of a quantum register. Another relevant design has been described by D. A. Lidar and L.-A. Wu in “Reducing Constraints on Quantum Computer Design by Encoded Selective Recoupling,” Physical Review Letters, Vol. 88, 017905 (2002), on the web as ArXiv.org/abs/quant-ph/0109021 (2001). The described encoding schemes, however, were not applicable directly to superconducting qubits.
In general, single qubit biasing often requires external circuitry or even off-substrate systems that are difficult to design and fabricate. In addition, the physical interactions driving the single qubit operations are much slower than those used in two qubit operations. Promising solutions can be developed by replacing one-qubit operations by two-qubit operations. Such designs eliminate the slow and hard-to-fabricate external circuitry, leading to a more simplified implementation as well as to increases in the speed of operations, and other benefits.
To achieve these goals, physical qubits are replaced by logical qubits that are formed from two or more physical qubits; this is called encoding. Second, the single qubit operations are performed through the cooperation of the two or more physical qubits that make up the logical qubit. Some early proposals of such encoding schemes have been put forward, for example, by D. Bacon, J. Kempe, D. P. DiVincenzo, D. A. Lidar, and K. B. Whaley in “Encoded universality in physical implementations of a quantum computer,” Proceedings of the 1st International Conference on Experimental Implementations of Quantum Computation (ed. R. G. Clark), pp. 257-264, (Rinton, N.J., 2001), available at ArXiv.org/abs/quant-ph/0102140, incorporated herein by reference in its entirety. Bacon et al. describe a logical qubit encoded via a grouping of physical qubits via Heisenberg interactions in a microscopic quantum computer.
Recently, the concept of recoupling of encoded states was proposed by D. A. Lidar and L.-A. Wu, “Reducing Constraints on Quantum Computer Design by Encoded Selective Recoupling”, Phys. Rev. Lett., 88:017905 (2002), available as ArXiv.org/abs/quant-ph/0109021 (2001), incorporated herein by reference in its entirety. However, the systems described differ in essential ways from the designs of superconducting quantum computers. Recoupling with pulses has been used in NMR based quantum information processing.
To date, no proposal has been put forward for a superconducting quantum computer that does not require individual qubit biasing. Nor is there a proposal for a superconducting quantum computer with encoded qubits. Finally, no proposals exist for operating such superconducting quantum computers efficiently.
Therefore, there is a need for superconducting quantum computers that replace single qubit operations, such as biasing, by encoded two qubit operations, while retaining the universality of the quantum computer. Fast and efficient methods of operating such quantum computers are also needed.